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Abstract

A model with two roughness levels for the diffraction of a plane wave by a metallic grating with periodic imperfections is presented. The grating surface is the sum of a reference profile and a perturbation profile. First, the diffraction by the reference grating is treated. At this stage the Chandezon method is used. This method leads to the resolution of eigenvalue systems. Each eigensolution defines an elementary wave function that characterizes a propagating or an evanescent wave. Second, the periodic errors are taken into account and a Rayleigh hypothesis is expressed: Everywhere in space the diffracted fields can be written as a linear combination of reference wave functions. The boundary conditions on the perturbed grating allow the diffraction amplitudes to be determined and therefore lead to the energetic magnitudes (efficiencies). The domain of analytical validity of this hypothesis is not defined. In fact, this method is considered to be an approximation. The proposed numerical study leads to some utilization rules. With a plane as the reference surface, the electromagnetic fields are given by classical Rayleigh expansions. Here the reference profile is a grating, hence the term generalized Rayleigh expansion.

References

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a Here a(x)=a0(x)+a1(x), with a0(x)=0.4λcos[(2πx/D0)-10°]+0.1λcos[(πx/D0)-10°],a1(x)=n0λ/40cos(2πx/n0D0), and λ/D0=1,θ=20°,ν(1)=1,v(2)=0.07624-1.431j. The accuracy Δξn(1)>A is obtained from M>MC,n0. The CPU times TC,n0=TC1,n0+TC2,n0 in seconds correspond to computing diffraction efficiencies with the matlab program on a Power-Mac 8500. The values TC2,n0 take into account only the E‖ polarization. For the H‖ case with the same accuracy, the truncation order MC,n0 and the computation times are different.

Table 2

Spectra of Elementary Wave Functions Associated with n0 Incidences

Real Incidence: sinθ and p=0

n0-1AdditionalIncidences:sinθ+pλ/D,p≠0

Spectrum of 4M+2 elements

n0-1 spectra of 4M elements

ψ0,n(j)(x,u)=ϕ0,n(j)(x)exp(jk(j)r0,n(j)u)

ψp,n(j)(x,u)=ϕp,n(j)(x)exp(jk(j)rp,n(j)u)

2M+1 outgoing waves

2M outgoing waves

2M+1 incoming waves

2M incoming waves

Table 3

Accuracy in Efficiencies versus Grating Period with the Generalized Rayleigh Expansiona

Accuracy

Grating Number n0

1

2

3

4

5

Δξn(1)>1

M>MR,1=4

M>MR,2=6

M>MR,3=5

M>MR,4=11

M>MR,5=10

∑TR=17.9

TR1,1=0.2

TR1,2=0.7

TR1,3=0.7

TR1,4=4.4

TR1,5=4.4

TR2,1=0.2

TR2,2=0.5

TR2,3=0.5

TR2,4=5.1

TR2,5=7.2

TR,1=0.4

TR,2=1.2

TR,3=1.2

TR,4=9.5

TR,5=11.6

Δξn(1)>2

M>MR,1=7

M>MR,2=8

M>MR,3=7

M>MR,4=11

M>MR,5=14

∑TR=37.7

TR1,1=0.3

TR1,2=0.9

TR1,3=2.1

TR1,4=4.4

TR1,5=11

TR2,1=0.3

TR2,2=0.7

TR2,3=1.0

TR2,4=5.2

TR2,5=19.5

TR,1=0.7

TR,2=1.6

TR,3=3.1

TR,4=9.6

TR,5=30.5

Δξn(1)>3

M>MR,1=8

M>MR,2=11

M>MR,3=10

M>MR,4=16

M>MR,5=18

∑TR=73

TR1,1=0.3

TR1,2=3

TR1,3=2.4

TR1,4=12.5

TR1,5=19.8

TR2,1=0.3

TR2,2=1.1

TR2,3=1.8

TR2,4=15.1

TR2,5=34.9

TR,1=0.6

TR,2=4.1

TR,3=4.2

TR,4=27.6

TR,5=54.7

Δξn(1)>4

M>MR,1=10

M>MR,2=14

M>MR,3=13

M>MR,4=16

M>MR,5=21

∑TR=99.6

TR1,1=0.9

TR1,2=3.8

TR1,3=5.1

TR1,4=12.5

TR1,5=35.2

TR2,1=0.8

TR2,2=3.5

TR2,3=3.8

TR2,4=15.1

TR2,5=41.2

TR,1=1.7

TR,2=7.3

TR,3=8.9

TR,4=27.6

TR,5=76.4

a Parameters and comments are as for Table 1. It should be observed that the ideal case MC,n0=n0MR,n0 is not the general case.

Table 4

Norm Values σc and σd for the Gratings Studied in Subsection 5.A

Norm Value

Amplitude

h1=0.02λ

h1=0.04λ

h1=0.08λ

h1=0.12λ

σc

0.1331

0.2661

0.5024

0.6892

σd

0.1351

0.2662

0.5026

0.6896

Class of Results

1 in E‖

1 in E‖

1 in E‖

1 in E‖

1 in H‖

1 in H‖

2 in H‖

2 in H‖

Table 5

Truncation Order Range ΔM Ensuring the Exploitation of Diffraction Efficiencies for the Metallic Grating Referenced in Subsection 5.D

a Here a(x)=a0(x)+a1(x), with a0(x)=0.4λcos[(2πx/D0)-10°]+0.1λcos[(πx/D0)-10°],a1(x)=n0λ/40cos(2πx/n0D0), and λ/D0=1,θ=20°,ν(1)=1,v(2)=0.07624-1.431j. The accuracy Δξn(1)>A is obtained from M>MC,n0. The CPU times TC,n0=TC1,n0+TC2,n0 in seconds correspond to computing diffraction efficiencies with the matlab program on a Power-Mac 8500. The values TC2,n0 take into account only the E‖ polarization. For the H‖ case with the same accuracy, the truncation order MC,n0 and the computation times are different.

Table 2

Spectra of Elementary Wave Functions Associated with n0 Incidences

Real Incidence: sinθ and p=0

n0-1AdditionalIncidences:sinθ+pλ/D,p≠0

Spectrum of 4M+2 elements

n0-1 spectra of 4M elements

ψ0,n(j)(x,u)=ϕ0,n(j)(x)exp(jk(j)r0,n(j)u)

ψp,n(j)(x,u)=ϕp,n(j)(x)exp(jk(j)rp,n(j)u)

2M+1 outgoing waves

2M outgoing waves

2M+1 incoming waves

2M incoming waves

Table 3

Accuracy in Efficiencies versus Grating Period with the Generalized Rayleigh Expansiona

Accuracy

Grating Number n0

1

2

3

4

5

Δξn(1)>1

M>MR,1=4

M>MR,2=6

M>MR,3=5

M>MR,4=11

M>MR,5=10

∑TR=17.9

TR1,1=0.2

TR1,2=0.7

TR1,3=0.7

TR1,4=4.4

TR1,5=4.4

TR2,1=0.2

TR2,2=0.5

TR2,3=0.5

TR2,4=5.1

TR2,5=7.2

TR,1=0.4

TR,2=1.2

TR,3=1.2

TR,4=9.5

TR,5=11.6

Δξn(1)>2

M>MR,1=7

M>MR,2=8

M>MR,3=7

M>MR,4=11

M>MR,5=14

∑TR=37.7

TR1,1=0.3

TR1,2=0.9

TR1,3=2.1

TR1,4=4.4

TR1,5=11

TR2,1=0.3

TR2,2=0.7

TR2,3=1.0

TR2,4=5.2

TR2,5=19.5

TR,1=0.7

TR,2=1.6

TR,3=3.1

TR,4=9.6

TR,5=30.5

Δξn(1)>3

M>MR,1=8

M>MR,2=11

M>MR,3=10

M>MR,4=16

M>MR,5=18

∑TR=73

TR1,1=0.3

TR1,2=3

TR1,3=2.4

TR1,4=12.5

TR1,5=19.8

TR2,1=0.3

TR2,2=1.1

TR2,3=1.8

TR2,4=15.1

TR2,5=34.9

TR,1=0.6

TR,2=4.1

TR,3=4.2

TR,4=27.6

TR,5=54.7

Δξn(1)>4

M>MR,1=10

M>MR,2=14

M>MR,3=13

M>MR,4=16

M>MR,5=21

∑TR=99.6

TR1,1=0.9

TR1,2=3.8

TR1,3=5.1

TR1,4=12.5

TR1,5=35.2

TR2,1=0.8

TR2,2=3.5

TR2,3=3.8

TR2,4=15.1

TR2,5=41.2

TR,1=1.7

TR,2=7.3

TR,3=8.9

TR,4=27.6

TR,5=76.4

a Parameters and comments are as for Table 1. It should be observed that the ideal case MC,n0=n0MR,n0 is not the general case.

Table 4

Norm Values σc and σd for the Gratings Studied in Subsection 5.A

Norm Value

Amplitude

h1=0.02λ

h1=0.04λ

h1=0.08λ

h1=0.12λ

σc

0.1331

0.2661

0.5024

0.6892

σd

0.1351

0.2662

0.5026

0.6896

Class of Results

1 in E‖

1 in E‖

1 in E‖

1 in E‖

1 in H‖

1 in H‖

2 in H‖

2 in H‖

Table 5

Truncation Order Range ΔM Ensuring the Exploitation of Diffraction Efficiencies for the Metallic Grating Referenced in Subsection 5.D